The present invention relates to a new and distinct variety of flat peach tree, Prunus persica, which has been given the variety denomination ‘ASFPBF0796’. This new tree produces fruit with a long shelf life without alteration both on the tree after growth completion and after harvesting, very good eating quality, semi-clingstone white flesh fruit for fresh market in July in the Pyrénées-Orientales departement, France. Contrast is made to ‘Nectarmagie’ (U.S. Plant Pat. No. 17,579) white nectarine tree, ‘ASFPBF0492’ (U.S. Plant Pat. No. 21,143) white flat peach tree, ‘Flatpretty’ (U.S. Plant Pat. No. 21,389) white flat peach tree, standard varieties, for reliable description. ‘ASFPBF0796’ is a promising candidate for commercial success in that it has very attractive fruits with very long shelf life without alteration before after harvesting.